Dual families of interacting particle systems on graphs
The author studies the general theory of duality for interacting particle systems (IPS) including many known special cases such as contact process, voter model, annihilating random walk (ARW), branching ARW, annihilating branching process (ABP), and biased ABP, etc. The first rule is about a simple condition for IPS with nearest neighbor interactions to be self-dual. It says: ``Every process with the contact transition, birth rate \(b > 0\) and no spontaneous birth is self-dual with self-duality parameter \(x =\) \(( d - a - c)/b\).'' Here \(a\) (resp. \(c\) or \(d\)) means annihilating (coalescence or death) rate, respectively. The next theorem says: ``There exist families of IPS \(\{ \alpha_i \}\), such that each \(\alpha_i\) is self-dual and dual to every other member of the family.'' Many of the most interesting processes have \(b > 0\). These processes either die out or the number of occupied sites tends to infinity, depending on some critical value \(b_c\) of birth rate \(b\). While, \(s_A\) denotes the survival probability that the set of 1's survives for all time when the initial set \(A\) is finite. When \(b < b_c\), \(s_A =0\), or when \(b > b_c\), \(0 < s_A \leq 1\), it is proved that all processes with \(b > 0\) possess a stationary measure \(\mu_{\infty}\) expressible as a function of \(s_A\), and the self-duality parameter \(x\). If extinction is certain for all finite initial sets \(A\), then \(\mu_{\infty}\) turns out to be \(\delta_{\varphi}\), which is always an equilibrium measure for any IPS without spontaneous birth. The theorem asserts: ``If an IPS on \({\mathbb Z}^k\) has self-duality parameter \(x \geq -1\), then it converges to \(\mu_{\infty}\) from any essentially infinite initial measure.'' The leading ideas and basic concepts such as duality and thinning for proofs of theorems in this paper are greatly due to the author and \textit{P. Lloyd} [Ann. Probab. 23, No. 4, 1816-1830 (1995; Zbl 0853.60079) and ibid 25, No. 1, 96-114 (1997; Zbl 0873.60075)]. As to other related works, see the author [Stochastic Processes Appl. 68, No. 2, 255-264 (1997; Zbl 0911.60082) and J. Phys. A, Math. Gen. 31, 8323-8331 (1998)] and also \textit{V. Belitsky, P. A. Ferrari, N. Kouno} and \textit{T. M. Liggett} [Stochastic Processes Appl. 67, No. 2, 213-225 (1997; Zbl 0890.60094)].
scientific article; zbMATH DE number 1552706